Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws☆

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Abstract

This paper contains a proof of the existence and uniqueness of solutions to the Riemann problem for systems of two hyperbolic conservation laws in one space variable. Our main assumptions are that the system is strictly hyperbolic and genuinely nonlinear. We also require that the system satisfy standard conditions on the second Fréchet derivatives, and one other hypothesis, which we have called the half-plane condition. This hypothesis replaces other, more restrictive hypotheses required by previous authors. The methods and results of this paper are designed to be applicable to systems of conservation laws which are not strictly hyperbolic.

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The research of the first author was partially supported by the National Science Foundation under Grant MCS 76-07654 and that of the second author by the United States Energy Research and Development Administration under Contract E(11-1)2456 with Columbia University.

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Present address: Dept. of Aerospace & Mechanical Sciences, Princeton University, Princeton, N.J. 08540.

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